Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II. Systems with a linear Poisson tensor

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $\ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $\ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $\ell(x,y)=c$. Set $B_0=\tfrac{1}{2}(B_1+B_3)$ and $B_5=\tfrac{1}{2}(B_2+B_4)$; these points lie on the line $\ell(x,y)=0$. Finally, let $B_\infty$ be the point at infinity on this line. Let $\mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_\infty$-switch and of the $B_0$-switch on the pencil $\mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=\ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $\Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $\Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.


INTRODUCTION
The Kahan discretization was introduced in [6] as a method applicable to any system of ordinary differential equations on R n with a quadratic vector field: where each component of Q : R n → R n is a quadratic form, while B ∈ Mat n×n (R) and c ∈ R n . Kahan's discretization reads as where is the symmetric bilinear form corresponding to the quadratic form Q. Equation (2) is linear with respect to x and therefore defines a rational map x = Φ f (x, ε). Explicitly, one has where f (x) denotes the Jacobi matrix of f (x). Clearly, this map approximates the time ε shift along the solutions of the original differential system. Since equation (2) remains invariant under the interchange x ↔ x with the simultaneous sign inversion ε → −ε, one has the reversibility property In particular, the map f is birational.
We will always set ε = 1 (and drop ε from notations). One can restore ε by the simple . In [7,8,10] the authors undertook an extensive study of the properties of the Kahan's method when applied to integrable systems. It was demonstrated that, in an amazing number of cases, the method preserves integrability in the sense that the map Φ f (x, ε) possesses as many independent integrals of motion as the original systemẋ = f (x). Further remarkable geometric properties of the Kahan's method were discovered in [1][2][3].

Theorem 1. [3] Consider a Hamiltonian vector field
where : R 2 → R is a linear form, and the Hamilton function H : R n → R is a polynomial of degree 2, Then the map Φ f (x) possesses the following rational integral of motion: where C(x, y), D(x, y) are polynomials of degree 2 given by and γ 1 = a 2 2 − a 1 a 3 , γ 2 = a 3 a 2 4 + a 1 a 2 5 − 2a 2 a 4 a 5 . The level sets of the integral (6) are conics which form a linear system (a pencil). We assume that the curve C(x, y) = 0 is nonsingular. The second basis curve of the pencil, D(x, y) = 0, is reducible, and consists of two lines (x, y) = ±c, where c = γ −1/2 1 .
All curves E λ pass through the set of base points which is defined by C(x, y) = D(x, y) = 0. Generically, there are four base points, counted with multiplicities. For any point (x 0 , y 0 ) ∈ C 2 different from the base points, there is a unique curve E λ of the pencil such that (x 0 , y 0 ) ∈ E λ . It is defined by λ = H(x 0 , y 0 ) = C(x 0 , y 0 )/D(x 0 , y 0 ). The orbit of (x 0 , y 0 ) lies on E λ .
We recall the definition of involutions on conics and on pencils of conics.

Definition 1. [4, 5].
1) Consider a nonsingular conic E in C 2 , and a point B ∈ E . The B-switch on E is the map I E ,B : E → E defined as follows: for any P ∈ E , the point I E ,B (P) is the unique second intersection point of E with the line (BP).
2) Consider a pencil E = {E λ } of conics in C 2 . The B-switch on E is the birational map I E,B : C 2 C 2 defined as follows. For any P ∈ C 2 which is not a base point of E, let E λ be the unique curve of the pencil such that P ∈ E λ , and set I E,B (P) = I E λ ,B (P).
The following statement is established in [5] in the case when H(x, y) is a quadratic form (a homogeneous polynomial of degree 2), that is, when a 4 = a 5 = 0. Theorem 2. The Kahan map Φ f can be represented as a composition of two involutions on the pencil E: are two points on the line (x, y) = 0.
The following geometric relation between ingredients of the construction passed unnoticed in [5].
The main result we would like to add in the present paper is that Theorem 2 can be reversed. We remark that one can take B 0 to be any of the four points , so that one obtains in this manner four different Kahan maps (or, better, two different Kahan maps along with their respective inverse maps).

PROOF OF THEOREM 3
Proof. Due to the covariance of the Kahan discretization with respect to rotations, we can restrict ourselves to the case (x, y) = x, so that α = 1 and β = 0. The Hamiltonian vector field f = J∇H for the Hamilton function H(x, y) = 1 2 a 1 x 2 + a 2 xy + 1 2 a 3 y 2 + a 4 x + a 5 y is given in this case byẋ = ∂H/∂y = a 2 x 2 + a 3 xy + a 5 x, The Kahan discretization of this system is the map ( x, y) = Φ f (x, y) defined by the equations of motion which can be solved for x, y according to where As a result, where R, S and T are polynomials of degree 2. They are given by R(x, y) = x 1 + a 5 + (a 2 + a 2 a 5 − a 3 a 4 )x + a 3 y , T(x, y) = 1 − a 5 − (a 2 + a 2 a 5 − a 3 a 4 )x − a 3 y − 2(a 2 2 − a 1 a 3 )x 2 . Now one shows by a direct computation that the system of three equations R(x, y) = 0, S(x, y) = 0, T(x, y) = 0 the singular points of Φ f admits exactly three solutions: and where c = (a 2 2 − a 1 a 3 ) −1/2 . Changing the signs of all a k , we find the three singular points of Φ −1 f : and Equations (10) follow immediately. One also confirms by a direct computation that the polynomial C(x, y) vanishes at the points B 1 , B 2 , B 3 , B 4 .

PROOF OF THEOREM 4
We start with formulas for computing involutions on conics.

Corollary 1.
For a pencil E consisting of the conics Proof. It is sufficient to observe that the formulas for the map I E λ ,B ∞ do not depend on λ. Indeed, the shift u 1 → u 1 + λα 2 , u 2 → u 2 + 2λαβ, u 3 → u 3 + λβ 2 leaves formulas (19), (20) invariant. Moreover, these formulas do not involve u 6 at all.

Lemma 2.
Let E be a nonsingular conic in C 2 given by the equation Then the map I E ,B : P 1 = (x 1 , y 1 ) → P 2 = (x 2 , y 2 ) is given by Proof. Proceeding as before, we observe that in this case equation of the line (B 0 P 1 ) is y = µx + ν. Thus, for the second intersection point P 2 = (x 2 , y 2 ) of this line with E we get a quadratic equation (21), with the coefficients A i given by (22)-(24). We compute x 2 from the Vieta formula x 1 + x 2 = −A 1 /A 2 , which leads to (25). To compute y 2 , we observe that which leads to (26) after a short computation.
Proof of Theorem 4. We set Consider the pencil of conics with the base points B 1 , B 2 , B 3 , B 4 . It will have the form (8) with a certain quadratic polynomial C(x, y) and with D(x, y) = c 2 − x 2 . For a given (x 0 , y 0 ), we determine the curve of the pencil through this point by setting The equation of this curve E is as in (18). We set B 0 = 0, (ξ 2 + ξ 4 )/2 , and use formulas of Lemmas 1, 2 to compute I E ,B ∞ • I E ,B 0 . Notice that in the present case, α = 1, β = 0, the formulas of Lemma 1 simplify considerably and read The result of this computation is that I E ,B ∞ • I E ,B 0 is given by formulas (17) with certain quadratic polynomials R(x, y), S(x, y), T(x, y). It remains to check whether this map satisfies equations of motion (14)-(15), which reduces to a system of linear equations for the coefficients a k . A direct computation shows that this system admits a unique solution: This proves the theorem.

CONCLUSIONS
In [9], we proved an amazing characterization of integrable maps arising as Kahan discretizations of quadratic planar Hamiltonian vector fields with a constant Poisson tensor, in terms of the geometry of their set of invariant curves. Here, these results are extended to quadratic planar Hamiltonian vector fields with a linear Poisson tensor. Such a neat geometric characterization of Kahan discretizations is quite unexpected and surprizing and supports our belief expressed in [7,8] that Kahan-Hirota-Kimura discretizations will serve as a rich source of novel results concerning algebraic geometry of integrable birational maps. It will be desirable to find similar characterizations for further classes of integrable Kahan discretizations, in dimensions n > 2.